A MODEL OF FUZZY TOPOLOGICAL RELATIONS FOR SIMPLE SPATIAL OBJECTS IN GIS

The goal of this paper is to present a new model of fuzzy topological relations for simple spatial objects in Geographic Information Sciences (GIS). The concept of computational fuzzy topological space is applied to simple fuzzy objects to efficiently and more accurately solve fuzzy topological relations, extending and improving upon previous research in this area. Firstly, we propose a new definition for simple fuzzy line segments and simple fuzzy regions based on computational fuzzy topology. And then, we also propose a new model to compute fuzzy topological relations between simple spatial objects, an analysis of the new model exposes:(1) the topological relations of two simple crisp objects; (2) the topological relations between one simple crisp object and one simple fuzzy object; (3) the topological relations between two simple fuzzy objects. In the end, we have discussed some examples to demonstrate the validity of the new model, through an experiment and comparisons of existing models, we showed that the proposed method can make finer distinctions, as it is more expressive than the existing fuzzy A model of fuzzy topological relations for simple... Bol. Ciênc. Geod., sec. Artigos, Curitiba, v. 21, n 2, p.389-408, abr-jun, 2015. 390 models. Keyword:Topological Relation; Simple Spatial Objects; Fuzzy Topology; Model; GIS. Resumo: O objetivo deste artigo é apresentar um novo modelo de relações topológicas fuzzy para objetos espaciais simples em Sistemas de Informações Geográficas (SIG). Aplica-se o conceito de espaço topológico fuzzy a objetos fuzzy simples para resolver as relações topológicas de modo mais eficiente e acurado. Inicialmente é proposta uma nova definição para segmentos de linha fuzzy e para regiões fuzzy com base na topologia computacional. Em seguida propõe-se um novo modelo para calcular as relações topológicas fuzzy entre os objetos espaciais. A análise do novo modelo aponta: (1) relações topológicas de dois objetos “crisp”; (2) relações topológicas entre um objeto “crisp” e um objeto fuzzy; (3) relações topológicas entre dois objetos “crisp”. Finalmente, discutem-se alguns exemplos para demonstrar a validade do modelo, por meio de um experimento e comparações entre modelos existentes. É possível demonstrar que o método proposto pode realizar distinções mais precisas e á mais expressivo do que os modelos fuzzy existentes. Palavras-chave:Relações Topológicas; Objetos Espaciais Simples; Topologia Fuzzy; Modelo; GIS.


Introduction
Geographical information sciences (GIS) commonly deal with geographical phenomena modeled by crisp points, lines, and regions, features which are clearly defined or have crisp boundaries.However, geospatial data are always uncertain or fuzzy due to inaccurate data acquisition, incomplete representation, dynamic change, and the inherent fuzziness of geographical phenomena itself.In GIS, many studies have been devoted to modeling topological relations, specifically the modeling of fuzzy topological relations between simple spatial objects.Topology is a fundamental challenge when modeling the spatial relations in geospatial data that includes a mix of crisp, fuzzy and complex objects.Two mechanisms, the formalization and reasoning of topological relations, have become popular in recent years to gain knowledge about the relations between these objects at the conceptual and geometrical levels.These mechanisms have been widely used in spatial data query (Egenhofer, 1997, Clementini et al., 1994), spatial data mining (Clementini et al., 2000), evaluation of equivalence and similarity in a spatial scene (Paiva, 1998), and for consistency assessment of the topological relations of multi-resolution spatial databases (Egenhofer et al., 1994(Egenhofer et al., , 1995;;Du et al., 2008a, b).Diloet al. (2007) defined several types and operators for modeling spatial data systems to handle fuzzy information.Shi et al. (2010) and Liu et al. (2011) developed a new object extraction and classification method based on fuzzy topology.However, the fuzzy topological relationships themselves must be modeled due to the existence of indeterminate and fuzzy boundaries between spatial objects in GIS.Fuzzy topology theory can potentially be applied to the modeling of fuzzy topological relations among spatial objects.To date, many models have been designed to formalize fuzzy topological relations between simple spatial objects these models provide a framework to conceptually describe the topological relations between two regions and can be considered as an extension of the crisp case.They can be implemented on spatial databases at less cost than other uncertainty models and are useful when managing, storing, querying, and analyzing uncertain data.For example, Egenhofer and Franzosa (1991a, 1994, 1995) and Winter (2000) modeled the topological relations between two spatial regions in two dimensional space (2-D) based on the 4-intersection model and ordinary point set theory.Li et al. (1999), Long and Li (2013) produced a Voronoi-based 9-intersection model based on Voronoi diagrams.Cohn et al. (1996Cohn et al. ( , 1997) ) discovered forty-six topological relations between two regions with indeterminate boundaries based on Region Connection Calculus (RCC) theory (Randell et al., 1992).Clementini andDi Felice (1996a, b, 1997) used extended 9-intersection model to classify forty-four topological relations between simple regions with broad boundaries.The extended 9-intersection model substantially agrees with the RCC model, though the former removes two relations considered as invalid in the geographical environment.The extended 9-intersection model can be extended to represent topological relations between objects with different dimensions, like regions and lines, while the RCC model can only be applied to relations between regions.Tang and Kainz (2002), Tang et al. (2005), Tang (2004) applied fuzzy theory and a 9-intersection matrix and discovered forty-four topological relations between two simple fuzzy regions.Shi and Liu (2004) discussed fuzzy topological relations between fuzzy spatial objects based on the theory of fuzzy topology.Du et al. (2005a, b) proposed computational methods for fuzzy topological relations description, as well as a new fuzzy 9-intersection model.Liu andShi (2006, 2009), Shi and Liu(2007) defined a computational fuzzy topology to compute the interior, boundary, and exterior parts of spatial objects, and based on the definition, Liu and Shi(2009) proposed a computational 9-intersection model to compute the topological relations between simple fuzzy region, line segment and fuzzy points, but the model did not give the topological relations between two simple fuzzy regions, and did not compute the topological relations between one simple fuzzy spatial object and one simple crisp spatial object.To further investigate the application of fuzzy topological relations in GIS, on the basis of previous researches, this study develops a new model of describing the fuzzy topological relations for simple fuzzy objects.The new model not only computes the topological relations between simple crisp spatial objects, but also computes the topological relations between simple fuzzy spatial objects.The remainder of this paper is organized as follows.In section 2, some basic concepts of fuzzy topology, computational fuzzy topology and the definitions of simple fuzzy spatial objects in GIS are detailed; In the Section 3, the new definition of simple fuzzy spatial objects is presented, and a new model of fuzzy topological relations for simple fuzzy spatial objects is proposed; In Section 4, some examples are discussed to validate the proposed method.Finally, some conclusions are drawn in Section 5.

A Brief Summary of Computational Fuzzy Topology
In this section, coherent fuzzy topologies, induced by interior and closure operators (Liu andShi, 2006, 2009;Shi and Liu, 2007;Liu and Luo, 1997), are reviewed.Mathematically, point set topology is the fundamental theory for modeling topological relations between simple crisp spatial objects in GIS.By extension, fuzzy topology is a generalization of ordinary topology that introduces the concept of membership value and can be adopted for modeling topological relations between spatial objects with uncertainties.Zadeh (1965) introduced the concept of fuzzy sets, and fuzzy set theory.Fuzzy topology was further developed based on the fuzzy sets (Chang, 1968;Wong, 1974;Wu and Zheng, 1991;Liu and Luo, 1997).Liu andShi (2006, 2009), and Shi and Liu (2007) defined a computational fuzzy topology to compute the interior, boundary and exterior of spatial objects.The computation is based on two operators, the interior operator and the closure operator.Each interior operator corresponds to one fuzzy topology and that each closure operator also corresponds to one fuzzy topology (Liu and Luo, 1997).The research detailed in this paper extends this work by defining fuzzy spatial objects.However, it is important to review basic concepts in fuzzy set theory as well as simple fuzzy objects in GIS.

Basic Concepts
We focus on the two-dimensional Euclidean plane R 2 , with the usual distance and topology.Fuzzy topology is an extension of ordinary topology that fuses two structures, the order and topological structures.Fuzzy interiors, boundaries, and exteriors play an important role in the uncertain relations between GIS objects.In this section we first present the basic definitions for fuzzy sets, and then the definitions for fuzzy mapping.Definition 2.1 (Fuzzy subset).Let X be a non-empty ordinary set and I be the closed interval [0, 1].
An I-fuzzy subset on X is a mapping A  : I X  , i.e., the family of all the [0,1]-fuzzy or I-fuzzy subsets on X is just I X ; consisting of all the mappings from X to I. Here, I X is called an I-fuzzy space.X is called the carrier domain of each I-fuzzy subset in it, and I is called the value domain of each I-fuzzy subset on X. X I A  is called a crisp subset on X, if the image of the mapping is the subset of   I  1 , 0 .

Definition 2.2 (Rules of set relations).
Let A and B be fuzzy sets in X with membership , respectively.Then, ] for all x in X; ] for all x in X; for all x in X.


. is called an I-fuzzy topology on X, and ( X I , ) is called an I-fuzzy topological space (I-fts), if  satisfies the following conditions: ，where J is an index set, and Where, 0  means the empty set and 1  means the whole set X. A .The closure of A is defined as the meeting of all the closed subsets containing A; denoted by A .
Definition 2.5 (Fuzzy complement).For any fuzzy set A, we defined the complements of ; denoted by C A .
Definition 2.6 (Fuzzy boundary).The boundary of a fuzzy set A is defined as: Definition 2.7 (Closure operator).An operator  is a fuzzy closure operator if the following conditions are satisfied:


, for all Definition 2.8 (Interior operator).An operator  is a fuzzy interior operator if the following conditions are satisfied: , for all

Definition 2.9 (Interior and closure operators). For any fixed
, both operators, interior and closure, are defined as respectively, and can induce an I-fuzzy topology ( are the closed sets.The elements in   and , for all fuzzy sets A, i.e., the complement of the elements in the   closed set.Details on how these two operators can induce a coherent I-fuzzy topology are given in Liu and Shi (2006).
To study topological relations, it is essential to first understand the properties of fuzzy mapping, especially homeomorphic mapping since topological relations are invariant in homeomorphic mappings.The following section presents a number of definitions related to fuzzy mapping. Let , define and its I-fuzzy reverse mapping continuous, and open (Liu and Luo, 1997).One important theorem to check an I-fuzzy homeomorphism is that, as proved by Shi and Liu (2007).Let be I-fts's induced by an interior operator and closure operators.Then ) , ( ) , ( : is a bijective mapping.Meanwhile, for the topology induced by these two operators, when checking a homeomorphic map, we only have to check whether there is a one-to-one correspondence between the domain and range.

Fuzzy Simple Spatial Objects in GIS
Based on the definitions presented in section 2.1, Liu and Shi developed fuzzy definitions for the basic elements in GIS (Liu andShi, 2006, 2009;Shi and Liu, 2007), summarized here as follows: Definition 2.10 (fuzzy point, Figure 1(a)).An I-fuzzy point on X is an I-fuzzy subset Definition 2.11 (Simple fuzzy line, Figure 1(b)).A fuzzy subset in X is called a simple fuzzy line (L) if L is a supported connected line in the background topology (i.e., a crisp line in the background topology).Definition 2.12 (Simple fuzzy line segment, Figure 1(b)).The simple fuzzy line segment (L) is a fuzzy subset in X with: 1) for any On the basis of definitions for simple fuzzy points, line segments and regions, Shi and Liu (2007) provide an example of computing the interior, boundary, and exterior of spatial objects for different values, and the interior, boundary, and exterior of spatial objects were confirmed for each given value.Based on the fuzzy definitions, Liu and Shi (2009) proposed a new 3 3  integration model to compute the topological relations between simple fuzzy region, line segment and fuzzy points.The element ( ) of the new 3 3  integration model is the ratio of the area( or volume) of the meet of two fuzzy spatial objects in a join of two simple spatial object(here a join of two fuzzy objects means "union" of two fuzzy objects; a meet of two fuzzy objects means "intersection" of two fuzzy objects (Liu and Shi, 2009)).And it was difficult to change or transform the new 3 3  integration model to describe the topological relations between one simple crisp spatial object and one simple fuzzy spatial object.Based on existing related studies, in the next section, we will discuss the method of fuzzy topological relations for simple spatial objects.

A New Definition for Simple Fuzzy Spatial Objects
Based on section 2, we developed a new definition for a simple fuzzy spatial line segment and region by applying the definition presented in this section.On fuzzy topological space (Chang, 1968), the fuzzy point definition remains the same as Definition 2.10.Definition 2.14 (inner and outer boundary of simple fuzzy line segment, Figure 2 . So, a simple fuzzy line segment L for given can be written as: . So the fuzzy region A for given can be expressed as:  can be considered as two simple crisp regions.
:the innerboundary of L :the outerboundary of , and Based on these definitions, the next section will primarily focus on discussing the new model of fuzzy topological relations for simple spatial objects.

Objects
In this paper, we just discussed the topological relations between two simple fuzzy line segments, the topological relations one simple fuzzy line segment and one simple fuzzy region, and the topological relations between two simple fuzzy regions, as follows.

(I) Topological relations between two simple fuzzy line segments
For one simple fuzzy line segment L1 for given  (figure3 (a)), and the other simple fuzzy line segment L2 for given  (figure3 (b)).The topological relations between L1 and L2 can be computed by 4 4  intersection model as equation ( 1).

2) If
, L2 is a simple fuzzy line segment, and comprised of four components ,L1 is a simple fuzzy line segment, and comprised of four components . Thus, the topological relations between L1 and L2 can be computed by 4 4 intersection model as equation ( 1).

(II)Topological relations between one simple fuzzy line segment and one simple fuzzy region
For one simple fuzzy line segment L1 for a given  (figure4 (a)), there is one simple fuzzy region A1 for given  (figure4 (b)).The topological relations between L1 and A1 can be computed by 4 4  intersection model as equation (3).
:the innerboundary of L There are four different topological relations between L1 and A1, as follows: , and A is a simple crisp region.The equation (3) can be turned into 4-Intersection Model (4IM) (Egenhofer and Franzosa, 1991a) or 9-Intersection Model (9IM) (Egenhofer and Franzosa, 1991b), the topological relations between L and A are computed by 4IM or 9IM.
The topological relations between L and A can be computed by 4 4  intersection model as equation ( 3).

(III) Topological relations between two simple fuzzy regions
For one simple fuzzy region A1 for given  (figure5 (a)), the other simple fuzzy region A2 for given  (figure5 (b)).The topological relations betweenA1 and A2 can be computed by 4 4  intersection model as equation ( 6).
, and The topological relations betweenA1 and A2 can be computed  intersection model as equation ( 6).Through the above description, the equation( 1), ( 3) and ( 6)are equivalent, only replacing the elements of the 4 4  intersection model, the 4IM or 9IM, equation( 2), ( 4) ,( 5), ( 7) are only the new 4 4  intersection models'(equation( 1), ( 3), ( 6)) exception, and all the equation can describe the topological relations respectively.In this section, we develop a new 4 4  intersection model to compute the fuzzy topological relations between simple spatial objects.An analysis of the new model exposes: (1) the topological relations between two simple crisp line segments; (2) the topological relations between two simple fuzzy line segments; (3) the topological relations between one simple crisp line segment and one simple fuzzy line segment; (4) the topological relations between one simple line segment and one simple crisp region; (5) the topological relations between one simple crisp line segment and one simple fuzzy region; (6) the topological relations between one simple fuzzy line segment and one simple crisp region; (7) the topological relations between one simple fuzzy line segment and one simple fuzzy region; (8) the topological relations between one simple crisp region and one simple fuzzy region; (9) the topological relations between two simple crisp regions; (10) the topological relations between two simple fuzzy regions.In the next section, we will focus on taking some examples to demonstrate the validity of the new model by comparing with existing models.

Experiment and Comparison
The new 4 4  intersection model can identify the topological relations between two simple spatial objects.The following content will take some examples to demonstrate the validity of the new model, and compare with the existing fuzzy models.

Experiment Results
In this section, we will take some examples to demonstrate the validity of the new model.1) A simple crisp line segment L1 (figure6 (a)), a simple fuzzy line segment L2 for given  =0.4 (figure6 (b)).The topological relation between them was shown in figure 6(c).Since the intersection of two sets can be either 0 or 1, the topological relation between L1 and L2 can be computed by equation (2) as: Bol  for given  =0.3 (figure7 (b)).The topological relation between them was shown in figure7(c).Since the intersection of two sets can be either 0 or 1, the topological relation between L1 and L2 can be computed by equation (1) as: .
:the innerboundary of L1 :the outerboundary of 3) A simple fuzzy line segment L1 for given =0.5 (figure8 (a)), a simple fuzzy region A1 for given  =0.4 (figure8 (b)).The topological relation between them was shown in figure8(c).Since the intersection of two sets can be either 0 or 1, the topological relation between L1 and A1 can be computed by equation (3) as:  4) as: :the outerboundary of L1 :the outerboundary of A1 The topological relation between L1 and A1 6) A simple fuzzy regionA1 for given =0.5 (figure11 (a)), a simple fuzzy region A2 for given  =0.3 (figure11 (b)), the topological relation between them was shown in figure11(c).

404
Since the intersection of two sets can be either 0 or 1, the topological relation between A1 and A2 can be computed by equation ( 6) as: 7) A simple fuzzy region A1 for given =0.5 (figure12 (a)), a simple crisp region A2 (figure12 (b)), the topological relation between them was shown in figure12(c).Since the intersection of two sets can be either 0 or 1, the topological relation between A1 and A2 can be computed by equation ( 7) as:

Comparison with Existing Models
In dealing with fuzzy spatial objects, Cohn and Gotts (1996) proposed the 'egg-yolk' model with two concentric sub-regions, indicating the degree of 'membership' in a vague/fuzzy region.In this model, the 'yolk' represents the precise part and 'egg' represents the vague/fuzzy part of a region.The 'egg-yolk' model is an extension of RCC theory into the vague/fuzzy region.A total of 46 relations can be identified (Cohn and Gotts, 1996).Based on the 9-intersection , which was proposed by Egenhofer and Franzosa (1991b), Clementini and Di Felice defined a region with a broad boundary, by using two simple regions (Clementini andDi Felice, 1996, 1997) , gave a total of 44 relations between two spatial regions with a broad boundary.For example, as shown in figure13(a, b), the extended 9-Intersection model proposed by Clementini andDi Felice (1996,1997) yielded the same matrix, , that is to say, the topological relations are same, but they are obviously different from each other as shown in figure13(a, b) .Meanwhile, Liu and Shi (2009) proposed a 3 3 integration model to compute the topological relations between fuzzy line segments, and discovered sixteen topological relations between simple fuzzy region and simple fuzzy line segment.However, the 3 3 integration model could not get the topological relation as shown in figure 13(c).Then, we will discuss the topological relation as shown in figure 13  Through the above comparison analysis, the new proposed model (when taking different values of and  ) not only can compute the topological relations as listed in existing studies (Liu and   Shi, 2009; Cohn et al. , 1996, 1997;Clementini and Di Felice,1996,1997), but also the topological relations not currently listed.

Conclusion and Discussion
Fuzzy topological relations between simple spatial objects can be used for fuzzy spatial queries and spatial analyses.This paper presented a model of fuzzy topological relations for simple spatial objects in GIS.Based on the research of Liu and Shi, we propose a new definition for simple fuzzy line segments and simple fuzzy regions based on the computational fuzzy topology.We also propose a new 4 4 intersection models to compute the fuzzy topological relations between simple spatial objects, as follows: (1) the topological relations of two simple crisp objects; (2) the topological relations between one simple crisp object and one simple fuzzy object; (3) the topological relations between two simple fuzzy objects.We have discussed some examples to demonstrate the validity of the new model.Through an experiment and comparisons of results, we showed that the proposed method can make finer distinctions, as it is more expressive than the existing fuzzy models.
In this study, fuzzy topology is dependent on the values of and  used in leveling cuts, and different values of  and  generate different fuzzy topologies and may have different topological structures.When some applications of fuzzy spatial analyses, an optimal value of and  can be obtained by investigating these fuzzy topologies (Liu and Shi, 2006, Shi and   Liu, 2007).
line  L is a supported connected lineLiu, B. et  al.Bol.Ciênc.Geod., sec.Artigos, Curitiba, v. 21, n o 2, p.389-408, abr-jun, 2015.395segment (i.e., a crisp line segment in the background topology) in the background topology sets in the background topological space.And, 2) in the background topological space, any outward normal from Supp (  A ) must meet Supp ( point (b) Fuzzy line segment (c) Fuzzy region Figure1: (a) Fuzzy point for a given ; (b) Fuzzy line segment for a given ; (c) Fuzzy region for a given  (Liu and Shi 2009) (a)), for given  , the interior and boundary of simple line segment L(as shown in figure1(b)) are confirmed, separately, can be regarded as a simple crisp line segment.Therefore, we define    L as the outer-boundary of L, and o L   as the inner-boundary of L, o L  as the interior of L, and  L  as the boundary of L(as shown in figure4(a)): o Fuzzy line segment L (b) Fuzzy region A Figure 2: (a) The inner-outer boundary of simple fuzzy line segment L for given ; (b) The inner-outer boundary of simple fuzzy region A for given Based on the above definitions, (1) to a simple fuzzy line segment L for given , if o (a) A simple fuzzy line segment L1 for given ; (b) A simple fuzzy line segmentL2

Figure 4 :
(a) A simple fuzzy line segment L for given ; (b)A simple fuzzy region A for given 

Figure 5 :
Figure 5: (a) A simple fuzzy region A1 for given ; (b) A simple fuzzy region A2 for given 